Host-pathogen coevolution and the corresponding phase diagram

Predicting the possible outcomes in a coevolutionary arms race between the predator and prey is the key to developing strategies that mitigate the risk of extinction of a species. For instance, in microbial systems the host will maintain resistance to the pathogen as long as the benefits outweigh the costs. Predicting this cost-benefit ratio in systems that coevolve is an open problem in the field. Here we present a simple three-dimensional model of host-pathogen interactions where the population dynamics can be analytically determined. In this system we describe the coevolution in terms of bifurcation theory, where changes in host/pathogen parameters lead to changes in population dynamics, from stable fixed points where resistance is maintained to periodic solutions to pathogen extinction. We recast these long-term outcomes in a phase diagram; trajectories in this diagram correspond to different coevolutionary pathways. This simple model can give insights into the evolutionary dynamics at a general level, extending to general host-pathogen and host-parasite systems, with possible applications in cancer biology.